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M. Teresa Balaguer-Coll, Diego Prior and Emili Tortosa-Ausina WP-EC 2010-02

Output complexity, environmental conditions, and the effi ciency of municipalities: a robust approach

Los documentos de trabajo del Ivie ofrecen un avance de los resultados de las investigaciones económicas en curso, con objeto de generar un proceso de discusión previo a su remisión a las revistas científicas. Al publicar este documento de trabajo, el Ivie no asume responsabilidad sobre su contenido. Ivie working papers offer in advance the results of economic research under way in order to encourage a discussion process before sending them to scientific journals for their final publication. Ivie’s decision to publish this working paper does not imply any responsibility for its content. La Serie EC, coordinada por Matilde Mas, está orientada a la aplicación de distintos instrumentos de análisis al estudio de problemas económicos concretos. Coordinated by Matilde Mas, the EC Series mainly includes applications of different analytical tools to the study of specific economic problems. Todos los documentos de trabajo están disponibles de forma gratuita en la web del Ivie http://www.ivie.es, así como las instrucciones para los autores que desean publicar en nuestras series. Working papers can be downloaded free of charge from the Ivie website http://www.ivie.es, as well as the instructions for authors who are interested in publishing in our series. Edita / Published by: Instituto Valenciano de Investigaciones Económicas, S.A. Depósito Legal / Legal Deposit no.: V- Impreso en España ( 2010) / Printed in Spain ( 2010)

WP-EC 2010-02

Output complexity, environmental conditions, and the efficiency of municipalities:

a robust approach*

M. Teresa Balaguer-Coll, Diego Prior and Emili Tortosa-Ausina**

Abstract Over the last few years, many studies have analyzed the productive efficiency of local governments in different countries. An accurate definition of their output bundles -i.e., the services and facilities they provide to their constituencies- is essential to this research. However, several difficulties emerge in this task. First, since in most cases the law only establishes the minimum amount of services and facilities to provide, it may well be the case that some municipalities go beyond the legal minimum and, consequently, might be labeled as inefficient when compared to other municipalities which stick to the legal minimum. Second, municipalities face very different environmental conditions, which raises some doubts about the plausibility of an unconditional analysis. This study tackles these problems by proposing a metafrontier analysis in which the efficiency of municipalities is evaluated after splitting them into clusters according to various criteria (output mix, environmental conditions, size). We perform our estimations using order-m frontiers, given their robustness to outliers and immunity to the curse of dimensionality. We provide an application to Spanish municipalities, and results show that both output mix and, more especially, environmental conditions, should be controlled for, since efficiency differences between municipalities in different groups are notable. Keywords: efficiency, environmental conditions, local government, metafrontier, order-m JEL Classification: D24, D60, H71, H72

Resumen Durante los últimos años muchos trabajos han venido analizando la eficiencia productiva de las corporaciones locales de una gran variedad de países. Para este tipo de estudios resulta crucial una definición precisa de los servicios e infraestructuras que los municipios proporcionan a sus ciudadanos. Sin embargo, esta tarea presenta varias dificultades. En primer lugar, dado que en muchas circunstancias la ley únicamente establece los servicios mínimos que debe proporcionar un municipio, puede darse el caso de que algunos municipios vayan más allá de este mínimo legal y, consecuentemente, sean clasificados como ineficientes al compararlos con otros municipios que se ciñen al mínimo. En segundo lugar, las corporaciones locales operan en condiciones ambientales muy dispares, lo cual genera dudas acerca de la factibilidad de un análisis incondicional. Este trabajo aborda estas cuestiones proponiendo un análisis metafrontera en el que la eficiencia de las corporaciones locales se evalúa tras clasificarlas en distintos grupos de acuerdo con criterios múltiples (output mix, condiciones ambientales, tamaño). Las estimaciones son llevadas a cabo utilizando fronteras orden-m, debido a la robustez que presentan frente a observaciones atípicas y la inmunidad a la “maldición de la dimensionalidad” (curse of dimensionality). Llevamos a cabo una aplicación a los municipios españoles, y los resultados indican que tanto el output mix como, sobre todo, las condiciones ambientales, deberían ser tenidas en cuenta al evaluar la eficiencia, pues las diferencias en la eficiencia de los municipios en los distintos grupos son notables. Palabras clave: eficiencia, condiciones ambientales, gobierno local, metafrontera, orden-m

* M.T. Balaguer-Coll and D. Prior acknowledge the financial support of the Ministerio de Educación y Ciencia (SEC2003-04770). E. Tortosa-Ausina acknowledges the financial support of Fundació Caixa Castelló-Bancaixa (P1.1B2008-46), and the Ministerio de Ciencia e Innovación (ECO2008-03813/ECON and ECO2008-05908-C02-01/ECON). ** M.T. Balaguer-Coll: Universitat Jaume I. D. Prior: Universitat Autònoma de Barcelona and IESEG School of Management. E. Tortosa-Ausina: Universitat Jaume I and Ivie. Contact author: tortosa@uji.es

  3

1. Introduction

Over the last few years, a wide range of studies has analyzed the productive efficiency of municipalities

from multiple perspectives. The empirical evidence now available is increasing, and can be divided into

two groups. On the one hand, some studies analyze efficiency in the provision of a specific service such as

refusal collection (Brueckner, 1981; Ruggiero, 1995; Bosch et al., 2000). However, in some countries this

type of study presents certain disadvantages related to the difficulties in assigning the amount of input

usage by each specific service. On the other hand, many studies have considered a global perspective,

taking into account that local governments provide their constituencies with a wide variety of services

and facilities from the municipal budget. The literature is fairly extensive, yet scattered in time—studies

have not emerged abruptly but rather have appeared sporadically over time.1

Therefore, this is a relevant topic whose importance has further increased in the last few years in some

countries. From a viewpoint of European integration and the Stability and Growth Pact (SGP), it is of

utmost importance that all layers of government manage their resources efficiently, in order to facilitate

and maintain European Economic and Monetary Union. The interest is even higher in certain European

countries such as Spain, where municipalities face tighter budget constraints since the passing of the law

on budget stability (“Ley General de Estabilidad Presupuestaria”, 2001), which establishes mechanisms to

control public debt and public spending in pursuit of a balanced budget.

A shared problem faced by the second category of studies referred to in the first paragraph is the

difficulty to accurately define, and measure, what it is that local governments produce. In most cases the

problems arise due to the impossibility of directly quantifying the supply of public services. The Spanish

case is not free from that criticism, although its magnitude is lessened, largely due to the availability

of data on most of the public services that municipalities are bound to provide, which also includes

information on output quality. However, the law only establishes the minimum services and facilities each

municipality must provide, which varies according to their population. Nothing prevents a particular

municipality from going beyond this legal minimum and providing not only more of each compulsory

service (such as discretionally increasing the area of public parks), but also providing additional services

and facilities whose input usage may be substantial. Such a municipality would be mislabeled as inefficient

when compared to other municipalities which stick to the legal minimum. Therefore, we can consider that

modelling municipalities’ output is difficult; we may even talk about output complexity (Haynes, 2003).

However, what is difficult to measure is the amount of services and facilities provided by each munici-

pality beyond the legal minimum. Some reports acknowledge this reality (see Vilalta and Mas, 2006), but

their results are generally based on surveys carried out on a limited number of municipalities. Previous

studies dealing with this reality are, for instance, Bennett and DiLorenzo (1982), Marlow and Joulfaian

(1989) or Merrifield (1994). According to some of these papers, the additional costs incurred by many

municipalities are quite large, although they also vary a great deal across observations. These studies1See, for instance, Brueckner and Wingler (1984), Deller (1992), Taylor (1995), De Borger and Kerstens (1996), Grossman

et al. (1999), Hughes and Edwards (2000) and, more recently, Sampaio de Sousa and Stošić (2005) Balaguer-Coll et al. (2007),or Barankay and Lockwood (2007).

2

constitute evidence supporting the hypothesis that some municipalities provide their constituencies with

a larger amount of services and facilities, not only because of different needs in different constituencies,

but also because of the different environmental conditions each municipality faces.

In addition to this, there is a large body of literature on the differing environmental conditions that

DMUs (Decision Making Units) face in different contexts—not only in local government. Since the pio-

neering contributions by Banker and Morey (1986a,b), many studies have analyzed the issue. Most of the

ensuing literature has been of an applied nature (see, for instance Bos and Kool, 2006), although several

theoretical refinements to the initial methodologies have also been proposed (see, for instance Ruggiero,

2004). This body of literature postulates that environmental conditions may have a strong impact on

DMUs’ performance. In the case of municipalities, this would imply that different local governments face

different constituencies in terms of economic and social conditions, to whose needs municipalities may be

more or less responsive irrespective of the amount of services they are obligated to provide. Municipalities

also have different characteristics in terms of geography (including, for instance, rugged terrain, or urban

sprawl); different sectoral production characteristics (in terms of tourism, etc.); and other characteristics.

We consider municipalities may provide more services and facilities than the legally established mini-

mum because of the different environmental conditions they face. Tourist municipalities may face budget

strains due to the highly increased personnel needs they have during their high season. Other municipal-

ities may face higher costs because of urban sprawl. According to Solé-Ollé and Hortas Rico (2008), the

urban spatial structure of many Spanish cities has not only an environmental impact, but also a major im-

pact on municipal finances. In other cases, reasons might be more involved, such as wealthy constituencies

asking for additional services, or rising expenditures on security because of the rapid population increases

experienced by some Spanish cities. As documented by Vilalta and Mas (2006), in a study applied to

a sample of the province of Barcelona, more than 30% of municipal expenditures were discretionary. In

these circumstances, one may reasonably expect that some municipalities will be mislabeled as inefficient

(or, at least, more inefficient than other municipalities) simply because of this wide variety of scenarios.

Therefore, it would be more appropriate to compare only local governments facing similar environmental

conditions.

This study approaches these problems through a two-stage analysis that firstly analyzes the productive

efficiency of municipalities, and secondly divides them into groups according to different classifications.

In the second stage we consider three criteria to classify municipalities into different groups. The first

one considers clusters according to local governments’ output mixes, in order to compare the efficiency of

municipalities with similar output bundles; in this way we can control for the fact that some of them might

provide services and facilities beyond the legal minimum—hence, more complex. The second criterion

constructs groups of municipalities for which we include information on environmental conditions. The

third criterion forms groups according to size, since the levels of services each municipality must provide

hinges on the level of population. In our application to Spanish municipalities, results show that both

output mix and the different environments that municipalities face are issues to control for, since the

3

differences between municipalities affiliated to different groups turned out to be statistically significant.

Therefore, in the first stage municipalities are classified into groups according to different criteria given

that, according to the hypotheses formulated, local governments in each group might be facing different

production opportunities. They respond by making choices from different sets of feasible input-output

combinations. These so-called technology sets differ because of differences in available stocks of physical,

human and financial capital (e.g., type of machinery, size and quality of the labor force, access to foreign

exchange), economic infrastructure (e.g., number of ports, access to markets), resource endowments (e.g.,

quality of soils, climate, energy resources) and any other characteristics of the physical, social and economic

environment in which production takes place. As indicated by O’Donnell et al. (2008), such differences

have led efficiency researchers to estimate separate production frontiers for different groups of DMUs.2

However, the literature on efficiency analysis has faced severe problems when dealing with the eval-

uation of DMUs in different groups, which are assumed to have different technologies. Authors such as

Battese and Rao (2002) or Battese et al. (2004) argue that the efficiencies of DMUs that operate under a

given production technology are not comparable with those of DMUs operating under different technolo-

gies. In our setting, this would imply that it is not possible to compare the efficiencies of municipalities

in the different groups formed according to our three criteria. Battese and Rao (2002) propose a solution

based on the concept of metafrontier3 in the context of efficiency measurement using stochastic frontier

analysis (SFA), which refined Battese et al. (2004). In their paper, Battese and Rao (2002) assume there

are two different data-generation mechanisms for the data—one with respect to the stochastic frontier,

the other with respect to the metafrontier model. In contrast, Battese et al. (2004) assume that the

metafrontier function is an overarching function that encompasses the deterministic components of the

stochastic frontier production functions for those DMUs operating under different technologies.

Yet in public sector applications such as the measurement of local government efficiency, most studies

have been using nonparametric techniques such as DEA (Data Envelopment Analysis), or its nonconvex

version (Free Disposable Hull, FDH), for a variety of reasons (Fox, 2001). O’Donnell et al. (2008) have

recently filled this gap in the literature, by extending the metafrontier to DEA and alternative SFA

approaches for estimating both metafrontiers and group frontiers. However, their solutions are not entirely

satisfactory, mainly because of the curse of dimensionality that generally affects efficiency scores obtained

using DEA. As indicated by Daraio and Simar (2007), increasing the number of inputs or outputs, or

decreasing the number of units being compared, leads to higher efficiencies, simply as a result of a statistical

artifact. Multiple applications in disparate fields are affected by this issue (see, among many others Maudos

et al., 2002), which is especially severe in fields such as mutual fund evaluation, where difficulties arise in

defining the number of inputs and outputs (Joro and Na, 2002). Our approach to deal with this problem

is based in the order-m frontier initially proposed by Cazals et al. (2002). We modify their algorithm2For example, separate frontiers have been estimated for universities in Canada (McMillan and Chan, 2006), Australia

(Worthington and Lee, 2008) and the United Kingdom (Glass et al., 1995), and for bank branches in South Africa (O’Donnelland Westhuizen, 2002) and Spain (Lovell and Pastor, 1997).

3The metafrontier function was first introduced by Hayami (1969) and Hayami and Ruttan (1970, 1971). As indicated bythese authors (Hayami and Ruttan, 1971, p. 82), “the metaproduction function can be regarded as the envelope of commonlyconceived neoclassical production functions.”

4

to control for the existence of municipalities facing different technologies (i.e., municipalities in different

groups), in such a way that the efficiencies found are now comparable because the curse of dimensionality

problem is strongly alleviated. Therefore, we contribute to this growing field of research in which, as

pointed out by Battese et al. (2004) in their conclusions, “further theoretical and applied studies with

other models for technical inefficiency effects are clearly desirable.”

The plan of the paper is as follows. Section 2 and Section 3 provide details on the techniques employed

to both measure efficiency and assess the statistical differences between the efficiencies obtained. Section

4 specifies the particularities of the data employed. Section 5 presents and comments on the most relevant

results, and finally Section 6 summarizes with some concluding remarks.

2. Methodology: efficiency measurement

In the first stage we measure efficiency for all municipalities in our sample, regardless of their different

characteristics. Therefore, we consider a common non-convex Free Disposable Hull (FDH) frontier for all

observations as follows:

min{αF DH

k ,Z1,Z2,...,ZS}αFDH

k ,

s.t. αFDHk TCk − ∑S

s=1 ZsTCs ≥ 0,∑Ss=1 Zsys,i − yk,i ≥ 0, i = 1, . . . , I

Zs ∈ {0, 1}, s = 1, . . . , S

(1)

where TCs is total cost for municipality s, s = 1, . . . , S, and ys,i represents the value of its ith output,

i = 1, . . . , I, and Zs denotes the intensity level at which the s observation is conducted.

The FDH methodology is particularly suited to detect the most obvious cases of inefficiency as this

technique is very demanding with regard to inefficiency measurement. For each municipality labeled as

FDH-inefficient, at least one other municipality with superior performance can be found in the sample.

Under some other technological assumptions (e.g., for the convex Data Envelopment Analysis, DEA,

models) it may well be the case that the inefficiency coefficient depends entirely on the assumption of

convexity.

At this point, two aspects of the FDH methodology deserve special attention: efficiency by default and

outliers. In the absence of a sufficient number of similar municipalities for a comparison, a municipality

is labeled as efficient by default. This ranking of efficiency does not result from any effective superiority,

but rather is due to the lack of information that would allow pertinent comparisons. In addition, by

construction, the FDH concept of efficiency applies both to the municipality that presents the lowest level

of spending and to those with the highest values for at least one output indicator. This extreme form of

the sparsity bias that characterizes the FDH technique leads to lack of discrimination among production

units and constitutes a shortcoming of the FDH approach.

As for outliers, by definition nonparametric frontiers are defined by the extreme values of the dimen-

5

sional space of inputs and outputs. Thus, the appearance of outliers (atypical observations that differ

significantly from the rest of the data) may considerably influence efficiency computations. It is therefore

necessary to verify that the divergence does not result from evaluation errors. However, once the reliability

of the data set has been confirmed this kind of information may provide valuable information.

Recent work has established the statistical properties of the FDH estimator (Kneip et al., 1998; Simar

and Wilson, 2000) so that inference is now possible either by using asymptotic results or by means of

bootstrap. Simar and Wilson (2000) present a survey on this issue as well as a detailed examination of the

statistical properties of the nonparametric estimators in a multivariate context. Like other nonparametric

measures, FDH estimators suffer from the curse of dimensionality due to their slow convergence rate.

Taken together, the above mentioned problems may be serious enough to jeopardize the FDH estimates.

To solve these problems some additional procedures are required in order to make FDH estimates more

robust. Several approaches have already been proposed in the literature. Wilson (1993, 1995) introduced

descriptive methods to detect influential observations in nonparametric efficiency calculations. More recent

developments include the order-m frontiers (Simar, 2003). The order-m approach, based on the concept

of expected maximal (minimal) output (input or cost) function, yields frontiers of varying degrees of

robustness. The order-m frontiers allow for statistical inference while keeping their nonparametric nature.

We briefly describe this approach below.

Consider a positive fixed integer m. For a given level of input (x0) and output (y0), the estimation

defines the expected value of maximum of m random variables (Y1,. . . ,Ym), drawn from the conditional

distribution of the output matrix Y observing the condition Ym ≥ y0. Formally, the proposed algorithm

(algorithm I) to compute the order-m estimator has the following steps:

1. For a given level of y0, draw a random sample of size m with replacement among those ysm, such

that ysm ≥ y0.

2. Compute Program (1) and estimate αs.

3. Repeat steps 1 and 2 B times and obtain B efficiency coefficients αbs(b = 1, 2, . . . , B). The quality

of the approximation can be tuned by increasing B, but in most applications B = 200 seems to be

a reasonable choice.

4. Compute the empirical mean of B samples as:

αms =

1B

B∑b=1

αbs (2)

As m increases, the number of observations considered in the estimation approaches the observed units

that meet the condition ysm ≥ y0 and the expected order-m estimator in each one of the b iterations (αbs)

tends toward the FDH (αFDHs ). Thus, m is an arbitrary positive integer value, but it is always convenient

to observe the fluctuations of the αbs coefficients depending on the level of m. For acceptable values of

6

m, αms will normally present values smaller than the unity (this indicates that these units are inefficient,

as total costs can be reduced without modifying the production plan). When αms > 1, the s unit can be

labeled as superefficient, as the order-m frontier exhibits a higher total cost.

As mentioned above, the order-m estimation is an excellent tool to mitigate the problems of dimension-

ality and the presence of extreme observations and outliers. However, this evaluation will be of little use

if part of the inefficiency found hinges on the output complexity or the different environmental conditions

that local governments face, which could lead to biased estimates of the frontier, and hence misleading

policy implications.

Therefore, our objective is to define a process that can estimate the impact on efficiency of output

complexity, environmental conditions and, in general, other options that consider classifying municipalities

in different groups. This estimation is possible following the proposals of Battese et al. (2004) and

O’Donnell et al. (2008) for estimating a metafrontier production function. This process (algorithm II)

contains the following steps:

1. Use cluster analysis to classify the S units in S1, S2, . . . , SC groups.

2. Following the algorithm to estimate the order-m efficiency coefficients, complete steps 1 to 4 of

algorithm I to estimate the efficiency coefficients (αm,S1s , αm,S2

s , . . . , αm,SCs ) for the municipalities

classified in each one of the clusters S1, S2, . . . , SC . In order to facilitate the cross-comparison of

the results, irrespective of the number of units classified in each cluster, the same value for m will

be assigned in all the estimations. By doing this, the problems of dimensionality and the potential

impact of the outliers will be neutralized.

3. After completing the conditional frontiers in step 2 of algorithm II, apply steps 1 to 4 of the order-

m estimation to the complete sample and estimate the efficiency coefficients with respect to the

metafrontier αms .

4. Estimate the technology gap ratio (TGR) separating the conditional and the metafrontier as the

ratio (αms /αm,S1

s ), (αms /αm,S2

s ), . . . , (αms /αm,SC

s ).

Figure 1 presents an illustration of a simple case with one output and the total cost. At a given output

level, the technology gap ratio (TGR) is defined as the lowest possible cost within the metafrontier divided

by the lowest total cost at the conditional specific cluster.4

3. Testing the closeness between efficiency distributions

Once computation of efficiency scores have been computed there are multiple ways to display and compare

results (El-Mahgary and Lahdelma, 1995). We consider some methods which provide us with more4We have borrowed the concept of technology gap ratio from Battese and Rao (2002), Battese et al. (2004) and O’Donnell

et al. (2008). They use the term technology because they consider DMUs operating under different technologies. Althoughwe also talk about different technologies, in order to facilitate comparisons with their papers, we must acknowledge we donot explicitly test whether the municipalities in the different groups have different technologies. Therefore, when referring totechnology we are only suggesting whether the effect of each hypothesis (output mix or environmental conditions) mattersor not.

7

Figure 1: Metafrontier function model

Cluster 01

Cluster 04

Cluster 02

Cluster 03

Total

cost

Output

8

accurate information (see, for instance Li, 1996; Li et al., 2009). If we based our interpretations on a

number of summary statistics only, we would miss a considerable amount of relevant information. Most

of these methods are based on kernel smoothing to nonparametrically estimate the density functions

corresponding to both αms and αm,SC

s indices.5 The kernel density estimator f for a univariate density f

based on a sample of S efficiency indices (either αms or αm,SC

s ) is f(x) = (Sh)−1∑S

s=1 K((x − αms )/h),

where s is the municipality index, αms represents its efficiency index, x is the evaluation point, h is the

bandwidth, and K is a kernel function satisfying certain properties. Additional decisions concerned the

kernel and bandwidth choice. For choice of kernel, we considered the Gaussian method for its ease of

computation. Regarding bandwidth selection, the plug-in method suggested by Wand and Jones (1994)

is increasingly adopted in the literature.6

We must control for the fact that efficiency indices are bounded between (0, 1]. The Silverman (1986)-

Schuster (1985) reflection method is based on the idea of “reflecting” the probability mass lying beyond

unity—where, theoretically, no probability mass should exist. The kernel estimate disregarding the bound-

ary condition can be shown to be biased and inconsistent (Simar and Wilson, 1998).7 After estimating

densities, the Li (1996) test shows whether the observed visual differences are statistically significant or

not. The test is based on measuring the distance between densities f(x) and g(x) for which we consider

the mean integrated square error, i.e.:

L = L(f(x), g(x)

)=

∫x

(f(x) − g(x)

)2dx =

∫x

(f2(x) + g2(x) − 2f(x)g(x)

)dx

=∫

x

(f(x)dF (x) + g(x)dG(x) − 2g(x)dF (x)

)(3)

where F and G are candidates for the distribution of X , with density functions f(x) and g(x) and f is

the nonparametric kernel estimator referred to above. Given that f = (Sh)−1∑S

s=1 K((x − αm

s )/h), and

g = (Sh)−1∑S

s=1 K((y − αm,SC

s )/h)

then, in practice, an estimator for L is:

L =∫

x

(f(x) − g(x)

)2dx

=1

S2h

S∑s=1

S∑t=1t�=s

[K

(αms − αm

t

h

)+ K

(αm,SCs − αm,SC

t

h

)− K

(αm,SCs − αm

t

h

)− K

(αms − αm,SC

t

h

)]

(4)

The integrated square error is also required for estimating the statistic on which the test is based, the5Several monographs cover this topic in depth. See, for instance, Silverman (1986), Pagan and Ullah (1999) or, more

recently, Li and Racine (2007).6It can also be easily accessed, as it is implemented in many software packages such as R.7Although we used a variety of statistical procedures to perform all computations, the FEAR package by Paul W. Wilson

for the statistical software R provides codes for computing both efficiencies and densities, considering the reflection method.See http://business.clemson.edu/Economic/faculty/wilson/.

9

expression for which is as follows:

T =Sh1/2L

σ(5)

where

σ =1

S2hπ1/2

S∑t=1

S∑s=1

[K

(αms − αm

t

h

)+ K

(αm,SCs − αm,SC

t

h

)+ 2K

(αms − αm,SC

t

h

)](6)

and h is the bandwidth.

The Li (1996) test requires some assumptions to be met such as independently distributed efficiency

scores in each sub-group (and identically within each sub-group). However, order-m efficiency estimates

are dependent in the statistical sense, since perturbations of observations lying on the estimated frontier

will in most cases cause changes in efficiencies estimated for other observations. As indicated by Simar

and Zelenyuk (2006), under these circumstances the Li (1996) test has to be modified in several ways.

First, we must control for the issue of bounded support in order-m density estimation. However, as

demonstrated by Simar and Zelenyuk (2006), although controlling for the boundary effect is important in

density estimation, the statistic based on the reflection method is “essentially the same as the original Li

(1996) test, with the difference being a factor of√

2 and the fact that the bandwidth used in estimation

of the statistic is obtained from the data with reflection rather than the original data”. Therefore, since

the independency issue is not negligible, we should then follow Simar and Zelenyuk (2006), who provide

a way of adapting the Li (1996) test to the order-m context via bootstrapping techniques to improve its

performance. These authors provide consistent bootstrap estimates of the p-values of the Li (1996) test

as follow:

p =1B

B∑b=1

I{Lb > L}, (7)

where b = 1, . . . , B is the number of bootstrap replicates. The p-values are then adapted to our context,

where the true efficiency scores are replaced by order-m estimates. Simar and Zelenyuk (2006) consider

somewhat ad hoc methods to solve the discontinuity problem generated by the spurious probability mass

at the unity—recall that, by construction, at least one observation will always be on the frontier, and in

most circumstances the number will be quite large. We adopt one of their proposed methods (Algorithm

II; see Simar and Zelenyuk, 2006), based on computing and bootstrapping the Li (1996) statistic using

the sample of order-m estimates where those equal to unity are “smoothed” away from the boundary. We

add a small noise, within, say, 5% of the empirical distribution of αms , disregarding those equal to the

unity, but with an order of magnitude smaller than the noise of the estimation. The smoothing procedure

is performed via:

αms =

⎧⎨⎩ αm

s + εs, if αms = 1;

αms , otherwise.

(8)

where εi = Uniform(0, min{S−2/(I+1+1), a}), a is the α-quantile of the empirical distribution of αms

ignoring those equal to the unity.

10

4. Data, inputs, and outputs

We perform the analysis for a sample of Spanish municipalities with a population over 1,000 inhabitants for

year 2000. Both input and output data are provided by the Spanish Ministry for Public Administration.

Information on outputs is gathered through the survey on local infrastructures and facilities (Encuesta de

Infraestructuras y Equipamientos Locales), which is performed with 5-year frequency and, consequently,

constrains our sample period. Information on inputs basically consists of different types of costs, and

is taken from local government budgetary data. This data is available for every year. The regions that

meet our criteria (data for year 2000, and data for both inputs and outputs) were Andalusia, Aragon,

Asturias, the Canary Islands, Cantabria, Castile-Leon, Castile-La Mancha, Extremadura, Murcia, La

Rioja, and the Valencian Community. The final sample was made up of 1,198 municipalities. There was

no information for the remaining regions for several reasons. At the time of the study, Madrid had not

yet presented information on its outputs. Catalonia, the Basque Country and Navarra do not have to

provide the Spanish Ministry for Public Administration with this information.

Measuring the production process at municipal level is usually more difficult than in other sec-

tors/industries. We can distinguish three stages in this process of transforming inputs into outputs

(Bradford et al., 1969). In the first stage primary inputs (labor, equipment and external services) are

transformed into intermediate outputs (e.g., hours of traffic control or the extension of police services).

In the second stage, intermediate outputs are transformed into direct outputs. This is what Bradford

et al. (1969) call D-outputs, which are ready for consumption by the population. In the third stage, the

direct outputs ultimately have welfare effects on consumers (e.g., increasing perceptions and feelings of

safety and welfare). The third stage of the process can be directly captured by outcome indicators (labeled

C-outputs by Bradford et al., 1969), which reflect the degree to which direct outputs translate into welfare

improvements as perceived by consumers.

The efficiency of municipalities can be measured at each stage of this production process. However,

under normal circumstances this will be difficult because data might be either unavailable or simply poor,

making it difficult to distinguish between primary inputs, intermediate outputs, direct outputs, and final

welfare effects. For this reason the analysis is usually confined to analyzing the first and second phases of

this process, i.e., the links between primary inputs and direct outputs. We base our selection of outputs

on the services and facilities provided by each municipality. All local authorities must provide public

street lighting, cemeteries, waste collection and street cleaning services, drinking water to households,

access to population centers, surfacing of public roads, and regulation of food and drink. In some cases

we must select proxies for these services and facilities. As pointed out by De Borger and Kerstens (1996),

population is assumed to proxy for the various administrative tasks undertaken by municipalities, but it

is clearly not a direct output of local production. Other relevant outputs, such as provision of primary

and secondary education, are not the responsibility of Spanish municipalities.

Spanish law requires municipalities to provide minimum services depending on their size. Some of the

minimum services and facilities must be provided by all municipalities, while others are only binding for

11

larger municipalities (with populations of over 5,000, 20,000, and 50,000, the boundaries that define the

different categories). The second column in Table 1 reports information on the minimum services that

each category of municipalities must provide. The third column indicates the selected output indicators

to measure the different services and facilities. Our output choice was driven by the minimum services

and facilities. The list of outputs for year 2000, along with summary statistics, are reported in Table 2.

The choice was also driven by previous studies on local government efficiency in other European countries,

since for the most part they are endowed with the same competencies.8

Selecting inputs is much easier, as it is based on budgetary variables reflecting municipalities’ costs. Our

definition reflects the economic structure of Spanish local government expenditures, details of which are

reported by Spanish legislation,9 that considers three basic categories: current or ordinary expenditures,

capital expenditures, and financial expenditures. Within these, current expenditures are further divided

into four chapters, or categories, which account for: i) personnel expenditure; ii) current goods and

services expenditures; iii) financial expenditures; iv) current transfers. Capital expenditures are also

broken down into either real investments, or capital transfers. The former is what Table 2 refers to as

capital expenditures (X4), i.e., all expenditures local governments implement: i) to produce or acquire

capital goods; ii) to acquire necessary goods to provide local services in the right conditions; or iii) financial

expenditures that are suitable for amortization. On the other hand, capital transfers (X5) refer to the

payments to institutions to finance certain investments. Descriptive statistics for year 2000 are provided

in Table 2. Since our analysis is entirely confined to overall cost efficiency, the fact that some local

government departments may be actually sharing some costs does not raise any particular issue.

We must also select those variables to include when carrying out the cluster analysis for classifying

municipalities into groups according to the different hypotheses. This task is not easy given that we face

certain relevant constraints. First, there is no well-established theory as to which variables constitute the

“environmental conditions” that might impact on each municipality’s cost structure. Second, the available

information is also limited. In a number of contexts the relevance of considering groups is unquestionable.

As indicated by O’Donnell et al. (2008), in most practical settings DMUs can be grouped a priori on

the basis of geographical, economic and/or political boundaries, to name a few.10 However, it is not

entirely clear how groups must be formed. In the absence of “natural” boundaries for the different groups,

multivariate statistical techniques such as cluster analysis are available for determining both the number of

groups and group membership. In our particular setting, the a priori classification or “natural” boundaries

would be those based on size, since the limits of each group are determined by the extent of their powers.

In contrast, we need the cluster analysis technique to classify municipalities according to either output

mix or environmental conditions.

Regarding the classification based on output complexity, the variables selected to construct the groups8Differences are basically confined to the area of education which, in Spain, corresponds to regional and central govern-

ments.9See Ministerial Order (Orden Ministerial), September 20th, 1989.

10As also indicated by O’Donnell et al. (2008), if the analysis were conducted in an SFA framework, it would also bepossible to conduct statistical tests concerning the number of groups. El-Gamal and Inanoglu (2005) propose other methodsto circumvent the use of multivariate analysis techniques.

12

Tab

le1:

Out

put

indi

cato

rsba

sed

onm

inim

umse

rvic

espr

ovid

edM

inim

umse

rvic

espr

ovid

edO

utpu

tin

dica

tors

All

loca

lgo

vern

men

ts

Pub

licst

reet

light

ing.

Num

ber

oflig

htin

gpo

ints

Cem

eter

yTot

alpo

pula

tion

Was

teco

llect

ion

Was

teco

llect

edSt

reet

clea

ning

Stre

etin

fras

truc

ture

surf

ace

area

Supp

lyof

drin

king

wat

erto

hous

ehol

dsPop

ulat

ion,

stre

etin

fras

truc

ture

surf

ace

area

Acc

ess

topo

pula

tion

cent

res

Stre

etin

fras

truc

ture

surf

ace

area

Surf

acin

gof

publ

icro

ads

Stre

etin

fras

truc

ture

surf

ace

area

Reg

ulat

ion

offo

odan

ddr

ink

Tot

alpo

pula

tion

Inlo

calgo

vern

men

tsw

ith

popu

lati

ons

ofov

er5,

000,

inad

diti

on

Pub

licpa

rks

Surf

ace

ofpu

blic

park

sP

ublic

libra

ryTot

alpo

pula

tion

,pu

blic

build

ings

surf

ace

area

Mar

ket

Mar

ket

surf

ace

area

Tre

atm

ent

ofco

llect

edw

aste

Was

teco

llect

ed

Inlo

calgo

vern

men

tsw

ith

popu

lati

onof

over

20,0

00,in

addi

tion

Civ

ilpr

otec

tion

Tot

alpo

pula

tion

Pro

visi

onof

soci

alse

rvic

esTot

alpo

pula

tion

,pu

blic

build

ings

surf

ace

area

,as

sist

ance

cent

ers

surf

ace

area

Fir

epr

even

tion

and

exti

ncti

onSt

reet

infr

astr

uctu

resu

rfac

ear

eaP

ublic

spor

tsfa

cilit

ies

Tot

alpo

pula

tion

,pu

blic

build

ings

surf

ace

area

Aba

ttoi

rTot

alpo

pula

tion

Inlo

calgo

vern

men

tsw

ith

popu

lati

ons

ofov

er50

,000

,in

addi

tion

Urb

anpa

ssen

ger

tran

spor

tse

rvic

eTot

alpo

pula

tion

,to

talsu

rfac

ear

eaP

rote

ctio

nof

the

envi

ronm

ent

Tot

alsu

rfac

ear

ea

13

Table 2: Summary statistics for inputs and outputs, year 2000Inputsa Mean Std.dev.

Wages and salaries (X1) 6,304.18 9,044.30Expenditure on goods and services (X2) 4,953.61 7,862.17Current transfers (X3) 1,072.51 1,748.62Capital expenditure (X4) 5,905.78 8,513.16Capital transfers (X5) 276.52 1,031.29

Outputs

Population (Y1) 6,118.54 6,991.48Number of lighting points (Y2) 1,100.07 1,022.89Tons of waste collected (Y3) 2,984.87 4,787.84Street infrastructure surface area (Y4) 256,097.13 261,653.68Public buildings surface areab(Y5) 599.84 1,198.27Market surface areab(Y6) 748.21 1,867.12Registered surface area of public parksb(Y7) 202.44 1,193.32Assistance centers surface area (Y8) 1,858.22 4,015.22

# of observations 1,198a In thousands of euros, converted from 1995 pesetas (1 euro=166.386 pese-

tas).b In square meters.

are similar to those chosen as outputs, dividing them by population. This helps to control for the fact

that, as pointed out in the introduction, some municipalities might go beyond the legal minimum and

provide an amount of services which does not correspond to their size. Therefore, we would only compare

municipalities with similar output mixes, i.e., only those in the same group. We have selected as many

variables to construct the clusters as outputs. The details are reported in Table 3.

Regarding the classification based on environmental conditions, we have chosen some of the variables

provided by the Anuario Estadístico de La Caixa.11 Although they can be partly judged as ad hoc, and

although factors influencing the amount, allocation and distribution of local public spending, we consider

these provide a rough idea of the environmental conditions facing each municipality that might have an

impact on municipal finances. Summary statistics are reported in Table 4, and the particular definition

of each variable considered for performing the cluster analysis follows:

Total surface area: (divided by population). This indicator is similar to density, which is usually defined

as urbanized land per person. Although some authors (Solé-Ollé and Hortas Rico, 2008) consider

it is more appropriate to use urbanized land per person, this variable was not available for all

municipalities in our sample. We consider this is an appropriate proxy for urban sprawl, although

other indicators can be considered (for instance, street surface area divided by total surface (ENV 1).

Tourist index: this index may contribute to rising municipal expenditures, at least during some months

of the year. Comparing these municipalities only with their peers might be more appropriate

(ENV 2).

Economic status: municipalities with wealthier populations might be facing higher types of require-

ments. These constituencies may be willing to pay more taxes but, in return, they will demand11An annual report provided by the largest Spanish savings bank, La Caixa.

14

Tab

le3:

Clu

ster

sba

sed

onou

tput

mix

,med

ians

,yea

r20

00

Gro

up#

obse

rvat

ions

Out

put

mix

1O

utpu

tm

ix2

Out

put

mix

3O

utpu

tm

ix4

Out

put

mix

5O

utpu

tm

ix6

Out

put

mix

7

Clu

ster

130

00.

0357

0.37

5556

.509

90.

0370

0.00

002.

2358

0.00

40C

lust

er2

407

0.02

040.

3825

30.1

844

0.03

150.

0638

2.44

040.

0083

Clu

ster

313

50.

0273

0.36

0143

.519

60.

0095

0.00

007.

5103

0.04

38C

lust

er4

116

0.05

470.

4442

108.

7945

0.12

670.

0000

2.84

840.

0000

Clu

ster

510

00.

0315

0.36

7065

.858

40.

5475

0.00

001.

6693

0.00

00C

lust

er6

720.

0234

0.89

2036

.413

40.

0410

0.04

062.

3416

0.01

02C

lust

er7

250.

0309

0.36

4348

.439

60.

0723

0.00

0045

.944

40.

0111

Clu

ster

814

0.03

032.

1255

46.8

649

0.03

240.

0075

3.97

910.

0093

Clu

ster

920

0.02

560.

3535

60.3

041

0.06

801.

0086

3.90

540.

0163

aT

heM

AN

OVA

anal

ysis

indi

cate

dth

atdi

ffere

nces

betw

een

the

diffe

rent

grou

pm

eans

wer

est

atis

tica

llysi

gnifi

cant

,as

show

nby

Wilk

s-Λ

=0.5

65,

corr

espo

ndin

gto

p-v

alue

=0.

000.

The

Pill

ai-B

artl

ett

stat

isti

cyi

elde

dan

alog

ous

resu

lts.

Out

put

mix

1:lig

htin

gpo

ints

/pop

ulat

ion

Out

put

mix

5:pu

blic

mar

kets

area

/pop

ulat

ion

Out

put

mix

2:w

aste

colle

cted

/pop

ulat

ion

Out

put

mix

6:pu

blic

park

sar

ea/p

opul

atio

nO

utpu

tm

ix3:

stre

etsu

rfac

ear

ea/p

opul

atio

nO

utpu

tm

ix7:

assi

stan

cece

nter

s/po

pula

tion

Out

put

mix

4:pu

blic

build

ings

surf

ace

area

/pop

ulat

ion

15

more services and facilities (ENV 3).

Industrial activities: (divided by population). Municipalities in which industrial activity is high may

be affected by a different, probably higher, cost structure, since this type of activity requires

higher investments in infrastructures, security, anti-pollution policies, etc., which may—although

not necessarily—be offset by higher tax revenues (ENV 4).

Total number of cars: (divided by population). This might be related to ENV 3. Some wealthy con-

stituencies have higher levels of education also (in relative terms) and might have a preference for

non-polluting means of transport (ENV 5).

Unemployment: higher unemployment might entail higher crime and, therefore, increased demands for

security (ENV 6).

Total population growth, 1991–98 (percentage): municipalities facing higher population growth might

have had to increase the services and facilities at their own expense, because of the speed at which

the population’s demands have increased. If central and regional governments have not reacted

promptly, they might face a sudden imbalance between their revenues and their costs (ENV 7).

Construction: (divided by population). Municipalities where construction was higher have also raised

high tax revenues, which might have led to inefficient management of these increased revenues. In

addition, the population levels in these municipalities might have increased sharply,12 which might

have driven some municipalities to increase their social expenditures, as well as other expenses such

as civil protection, security, etc., some of which are not included in the list of minimum services

municipalities must provide (ENV 8).

Agricultural vehicles/Total number of vehicles (percentage): this information would indicate whether

it is a rural municipality whose needs might differ from others with different sectoral specializations.

It might also be a proxy for urban sprawl (ENV 9).

Number of bank branches: (divided by total population). This would indicate another type of spe-

cialization and, in addition, it might proxy for the economic level of the municipality (ENV 10).

We admit the selection of these variables is somewhat ad hoc. However, we consider they represent a

realistic summary of the different socioeconomic conditions affecting each municipality.

Some decisions involved in performing cluster analysis are the measure of similarity as well as the

clustering method. Regarding the former, one of the most popular choices is the Euclidean square distance.

Regarding the latter, although there are several alternatives, the Ward hierarchial clustering method has

the advantage of maximizing intra-group homogeneity and inter-group heterogeneity. In addition, the

technique is robust to outliers and groups are not too dissimilar in size. However, the criterium that must12Indeed, the strong increase of population in Spain over the last 10 years has been unevenly distributed across regions,

and it has been especially higher in areas where construction grew more in relative terms.

16

Tab

le4:

Clu

ster

sba

sed

onen

viro

nmen

talv

aria

bles

,med

ians

,yea

r20

00a

Gro

up#

obse

rvat

ions

EN

V1

EN

V2

EN

V3

EN

V4

EN

V5

EN

V6

EN

V7

EN

V8

EN

V9

EN

V10

Clu

ster

144

433

.342

20.

0000

3.00

0012

.044

50.

4057

3.00

00–3

.600

05.

9265

3.30

900.

8857

Clu

ster

254

218.

3355

2.50

005.

0000

14.4

055

0.53

593.

6500

25.1

000

8.35

354.

4866

0.72

96C

lust

er3

9631

0.27

102.

0000

5.00

0013

.620

00.

5077

5.00

001.

8500

6.09

254.

1590

0.77

03C

lust

er4

201

69.8

400

1.00

004.

0000

20.7

760

0.50

633.

3000

–1.1

000

11.7

070

6.25

001.

0717

Clu

ster

513

099

.397

52.

0000

3.00

0011

.747

00.

4087

5.70

001.

8500

5.68

403.

8620

0.76

24C

lust

er6

147

20.7

421

1.00

005.

0000

24.5

230

0.49

882.

4000

–5.3

000

13.8

750

6.98

302.

1816

Clu

ster

771

75.1

765

1.00

004.

0000

16.4

570

0.53

743.

2000

2.60

008.

1300

19.5

360

0.96

43C

lust

er8

4410

8.61

352.

0000

5.00

0039

.999

50.

6188

2.75

0016

.150

019

.902

05.

1865

1.20

16a

The

MA

NO

VA

anal

ysis

indi

cate

dth

atdi

ffere

nces

betw

een

the

diffe

rent

grou

pm

eans

wer

est

atis

tica

llysi

gnifi

cant

,as

show

nby

Wilk

s-Λ

=0.5

02,co

rres

pond

ing

top-v

alue

=0.

000.

The

Pill

ai-B

artl

ett

stat

isti

cyi

elde

dan

alog

ous

resu

lts.

Res

ults

for

som

ecl

uste

rsar

eno

tre

port

eddu

eto

thei

rve

rysm

allnu

mbe

rof

obse

rvat

ions

,ba

sica

llyou

tlie

rsw

hich

coul

dno

tbe

clas

sifie

din

othe

rgr

oups

.T

hese

tota

lled

11ob

serv

atio

nsw

hich

,co

mpa

red

wit

hth

e1,

198

mun

icip

alit

ies

inou

rsa

mpl

e,is

not

note

wor

thy.

EN

V1:

surf

ace

area

/pop

ulat

ion

EN

V6:

unem

ploy

men

tra

teE

NV

2:to

uris

tin

dex

EN

V7:

popu

lati

ongr

owth

,19

91–9

8E

NV

3ec

onom

icst

atus

EN

V8:

cons

truc

tion

/pop

ulat

ion

EN

V4

indu

stri

alac

tivi

ties

/pop

ulat

ion

EN

V9:

agri

cult

ural

vehi

cles

/num

ber

ofve

hicl

esE

NV

5nu

mbe

rof

cars

/pop

ulat

ion

EN

V10

:nu

mbe

rof

bank

bran

ches

/pop

ulat

ion

17

Table 5: Order-m efficiencies (metafrontier), summary statistics, year 2000Group Mean Median Max. Min. Std.dev. % of eff. obs.

All municipalities, metafrontier 0.9118 1.0000 1.4207 0.2344 0.1818 0.3088Group 1, metafrontier 0.8904 1.0000 1.4207 0.2344 0.2048 0.1068Group 2, metafrontier 0.9417 1.0000 1.1364 0.3772 0.1343 0.5979Group 3, metafrontier 0.9787 1.0000 1.0000 0.6376 0.0730 0.9014

ultimately guide the decision on both the methodology and the optimal number of groups is whether the

final groups are sensible in any way, and whether statistical differences exist among group centroids.

Battese et al. (2004) point out the importance of analyzing whether all municipalities share the same

technology. If all municipality-level data were generated from a single production function and the same

underlying technology, there would be no good reason for estimating the efficiency levels of municipalities

relative to a metafrontier. We assume that if statistically significant differences existed among the different

groups, it would constitute evidence in favor of comparing municipalities with those in their group only.

5. Results

Table 5 provides summary statistics on unconditioned efficiency for order-m efficiency scores. Results

are reported for all municipalities, and also for the different size categories, given the differences in their

powers. Results have been obtained by specifying a common frontier—the metafrontier—for all 1,198

observations. Thus, although results are split into different municipality size categories, they correspond

to the same common frontier. The results corresponding to all municipalities, regardless of their powers,

are displayed in the first row. Average efficiency is 91.18%, which is a high value given that municipalities

would become fully efficient if they were able to decrease their total costs by 8.82% only. However, this is an

average effect which varies across municipalities. The values at both tails of the distribution suggest that a

remarkable variety of behaviors exist, since the minimum is 23.44%, whereas the maximum is 142.07%. In

the former case, cost inefficiency is high, whereas the latter refers to cases of super-efficiency—units which

lie beyond the frontier and can be regarded as outliers. This finding is important, since it constitutes a

clear advantage of the order-m frontiers over DEA or FDH, which are strongly affected by the existence

of outliers. In the case of order-m frontiers these extreme observations are labeled as super-efficient and

do not affect the efficiencies found for other observations.

Table 5 also reports order-m efficiencies for the different categories of municipalities split by population

and, consequently, levels of powers. The smallest municipalities in the sample (those with populations

between 1,000 and 5,000) are, on average, the most inefficient. Mean efficiency is 89.04%, close to the

global mean value of 91.18%. These results are partly similar because this is, by and large, the category

with more observations. In contrast, medium sized municipalities (with populations between 5,000 and

20,000) and large municipalities (with populations over 20,000) show higher efficiencies. Not only is

average efficiency higher (94.17% and 97.87%, respectively), but also the number of municipalities lying

18

on the frontier (i.e., fully efficient municipalities) is much higher (59.79% and 90.14%, respectively).13

However, the most interesting results are those obtained for the different clusters, constructed either

using output mix or environmental variables. A description of these clusters is provided in Table 3

(clusters based on output mix) and Table 4 (clusters based on environmental variables). In order to

facilitate interpretations, we provide a lower panel below each table that reports a summary of the variables

included to form the clusters. Regarding the clusters based on output mixes, as reported in Table 3,

differences between the municipalities in each group (in terms of the selected variables) are noteworthy,

even though the size of some of these clusters is remarkable. For instance, group 1 is made up of 300

municipalities, roughly 1/4 of the total sample. Ideally, it would be desirable to have clusters containing

fewer observations to facilitate comparisons. However, some clusters were difficult to split into further

groups, despite considering the Ward method to cluster observations (which tends to form equally-sized

clusters). This is an interesting finding, which would corroborate the fact that many municipalities indeed

do different things, making comparisons misleading.

In some cases, the medians for some clusters and variables differ substantially; this is the case for

cluster 2 in OUTMIX5, cluster 3 in OUTMIX6, cluster 4 in OUTMIX3, cluster 5 in OUTMIX4,

cluster 6 in OUTMIX2 and OUTMIX5, cluster 7 in OUTMIX6, cluster 8 in OUTMIX2, and cluster

9 in OUTMIX5. Therefore, the clusters excel in some particular variables, even taking into account that

some of them contain many observations—compared with the total sample size. In addition, although

there is a wide consensus that the multivariate technique of cluster analysis is flawed, especially because

of the multiple decisions it involves, the MANOVA analysis indicated that the differences between the

identified groups and variables were indeed significant.14

With respect to the clusters based on environmental variables, as indicated in Table 4, the differences

found among them (with respect to the variables included in the analysis) are also noteworthy, and

significant at the 1% level. In this case, differences are more difficult to distinguish, because of the narrow

range of variation for some of these variables (for instance, ENV 2 or ENV 3). However, certain groups

excel in some variables. See, for instance, cluster 3 in ENV 1, cluster 8 in ENV 4, cluster 2 and 6 in

ENV 7, cluster 8 in ENV 8, cluster 7 in ENV 9 or cluster 6 in ENV 10.

It may be claimed that the municipalities in the groups, regardless of the criteria followed to create

them, might be considered to possess different technologies. As indicated in the introduction, Battese

et al. (2004) propose a method for comparing the efficiencies of DMUs in different groups in the context

of Stochastic Frontier Analysis, which has been extended to the DEA context by O’Donnell et al. (2008).

Our methodology, based on order-m indicators, allows us to control for group membership in the context

of efficiency measurement via nonparametric techniques and at the same time, take into account the

severity of the curse of dimensionality. We report the results obtained following our approach in Table 6.

On average—and this result holds for both categories of clusters—municipalities are much closer to their13For explanations for the different levels of inefficiency found see, for instance, Balaguer-Coll et al. (2007).14Although, technically, MANOVA compares the means of the groups, not the medians, as reported in Table 3 and Table

4.

19

Table 6: Order-m efficiency scores and metatechnology ratios, all municipalities,summary statistics, year 2000

Group Index Mean Median Max. Min. Std.dev.

Clusters based onoutput mix variables

Cluster-specificefficiencies

0.9910 1.0000 1.9583 0.2425 0.1832

Technology gap 0.9249 0.9996 1.1949 0.3659 0.1343

Clusters based onenvironmentalvariables

Cluster-specificefficiencies

0.9892 1.0000 1.5218 0.2493 0.1447

Technology gap 0.9203 0.9895 1.1682 0.2766 0.1264

Clusters based on size Cluster-specificefficiencies

0.9264 1.0000 1.6983 0.2428 0.1807

Technology gap 0.9849 1.0000 1.2117 0.4559 0.0598

frontiers, as documented by average efficiencies closer to unity. This result holds for both clusters based

on output mix variables (99.10%) and clusters based on environmental variables (98.92%). In the case of

clusters based on size, as one might a priori expect, results are quite similar to those of the unconditioned

case (Table 5).

However, Table 6 also reveals that classifying municipalities into different groups does not per se

explain away the remaining efficiencies. The maximum values for the three clustering criteria are well

above the unity, suggesting that a non-negligible number of outliers exist. These are what Andersen and

Petersen (1993) call super-efficient units.15

More specific results are reported in Table 7, Table 8, and Table 9. They report basic summary

statistics of the technology gap ratio, the efficiencies obtained from the group frontiers (CEg), and the

metafrontier (CE∗). In the case of clusters based on output mix, the widest gap between group efficiencies

and metafrontier efficiencies corresponds to municipalities in cluster 2, for which average CEg2 = 1.0508

and average CE∗2 = 0.9071. As a result, the technology gap ratio is the lowest (TGR2 = 0.8776).

In contrast, for cluster 8 the technology gap ratio is the highest (TGR8 = 1.0009) which should be

interpreted inversely, i.e., the efficiencies for municipalities in this group are very similar for the group

frontier (CEg = 1.0006) and metafrontier (CE∗ = 1.0016). In general, although some groups show

remarkable discrepancies between group efficiencies and metafrontier efficiencies (clusters 1 and 2), for

many others the gap is narrower (well above 0.90). Although this result might constitute evidence against

our initial hypothesis, we should bear in mind that clusters 1 and 2 are indeed the largest ones, with 300

and 407 observations included in each of them. Therefore, for more than half of the municipalities in our

sample, it is more reasonable to compare them only with the municipalities in their output mix group.

Table 8 reports analogous information as in Table 7 for clusters constructed using environmental

variables. In this case, supporting evidence for our initial hypothesis is stronger as, on average, the

technology gap ratios are much lower than in Table 7. The widest gap is found for cluster 8, whose

average TGR8 = 0.8718, whereas the lowest gap is found for cluster 3 (TGR3 = 0.9421). However, in this

case the clusters are, on average, much closer to their respective group frontiers than in the case of output

mix clusters—on average, most of them show CEg values in the vicinity of 1. Therefore, regardless of the15The existence of outliers also partly underlies the remarkably high average values for the efficiency scores.

20

Table 7: Order-m efficiencies and metatechnology ratios, clusters based on output mix vari-ables

Group Mean Median Minimum Maximum Std.Dev. % eff. obs.

Group efficiency, CEg 0.9651 1.0000 0.3222 1.6823 0.1669Cluster 1 Technology gap ratio 0.9083 0.9678 0.4094 1.0466 0.1266 0.1900

Metafrontier, CE∗ 0.8809 1.0000 0.2672 1.4207 0.2047

Group efficiency, CEg 1.0508 1.0000 0.4205 1.9583 0.2278Cluster 2 Technology gap ratio 0.8776 0.9529 0.4089 1.0000 0.1489 0.3808

Metafrontier, CE∗ 0.9071 1.0000 0.3772 1.2286 0.1731

Group efficiency, CEg 0.9300 1.0000 0.2944 1.1426 0.1582Cluster 3 Technology gap ratio 0.9751 1.0000 0.4835 1.0384 0.0870 0.3037

Metafrontier, CE∗ 0.9090 1.0000 0.3049 1.1412 0.1795

Group efficiency, CEg 0.9796 1.0000 0.3445 1.2002 0.1043Cluster 4 Technology gap ratio 0.9785 1.0051 0.3659 1.1949 0.1272 0.1638

Metafrontier, CE∗ 0.9611 1.0040 0.2710 1.3073 0.1699

Group efficiency, CEg 0.9225 1.0000 0.2425 1.0874 0.1623Cluster 5 Technology gap ratio 0.9922 1.0006 0.4667 1.1652 0.0908 0.2300

Metafrontier, CE∗ 0.9165 1.0000 0.2536 1.1682 0.1854

Group efficiency, CEg 0.9972 1.0000 0.6334 1.1546 0.0749Cluster 6 Technology gap ratio 0.9418 1.0000 0.3846 1.0730 0.1252 0.4861

Metafrontier, CE∗ 0.9412 1.0000 0.3923 1.2117 0.1527

Group efficiency, CEg 0.9942 1.0000 0.9207 1.0191 0.0229Cluster 7 Technology gap ratio 0.9883 1.0000 0.7597 1.0145 0.0494 0.6000

Metafrontier, CE∗ 0.9832 1.0000 0.7076 1.0339 0.0619

Group efficiency, CEg 1.0006 1.0000 1.0000 1.0088 0.0023Cluster 8 Technology gap ratio 1.0009 1.0000 0.8551 1.1033 0.0502 0.5714

Metafrontier, CE∗ 1.0016 1.0000 0.8551 1.1129 0.0517

Group efficiency, CEg 0.9686 1.0000 0.3371 1.0187 0.1487Cluster 9 Technology gap ratio 0.9625 1.0000 0.6663 1.1369 0.1215 0.6500

Metafrontier, CE∗ 0.9411 1.0000 0.2344 1.1581 0.1963

21

Table 8: Order-m efficiencies and metatechnology ratios, clusters based on environmentalvariables

Group Mean Median Minimum Maximum Std.Dev. % eff. obs.

Group efficiency, CEg 1.0008 1.0000 0.3251 1.5098 0.1603Cluster 1 Technology gap ratio 0.9155 0.9573 0.4106 1.0131 0.1019 0.2297

Metafrontier, CE∗ 0.9183 1.0000 0.2672 1.3073 0.1775

Group efficiency, CEg 0.9923 1.0000 0.5929 1.0813 0.0615Cluster 2 Technology gap ratio 0.9200 1.0000 0.5185 1.0181 0.1381 0.4630

Metafrontier, CE∗ 0.9144 1.0000 0.4611 1.0503 0.1536

Group efficiency, CEg 0.9895 1.0000 0.4945 1.4171 0.1124Cluster 3 Technology gap ratio 0.9421 1.0000 0.2766 1.0003 0.1242 0.5833

Metafrontier, CE∗ 0.9326 1.0000 0.2768 1.2117 0.1588

Group efficiency, CEg 0.9811 1.0000 0.2493 1.5218 0.1863Cluster 4 Technology gap ratio 0.9358 0.9788 0.4822 1.0070 0.0949 0.3284

Metafrontier, CE∗ 0.9176 1.0000 0.2344 1.4207 0.1882

Group efficiency, CEg 1.0022 1.0000 0.4365 1.2965 0.1010Cluster 5 Technology gap ratio 0.9187 0.9985 0.3577 1.0045 0.1449 0.4231

Metafrontier, CE∗ 0.9209 1.0000 0.3049 1.1232 0.1658

Group efficiency, CEg 0.9557 1.0000 0.3818 1.2284 0.1382Cluster 6 Technology gap ratio 0.9190 1.0000 0.4005 1.1682 0.1547 0.1361

Metafrontier, CE∗ 0.8805 1.0000 0.3115 1.2286 0.2037

Group efficiency, CEg 0.9987 1.0000 0.8312 1.1401 0.0404Cluster 7 Technology gap ratio 0.9090 1.0000 0.3203 1.0829 0.1680 0.3944

Metafrontier, CE∗ 0.9095 1.0000 0.3203 1.1193 0.1756

Group efficiency, CEg 0.9655 1.0000 0.3860 1.0712 0.1303Cluster 8 Technology gap ratio 0.8718 0.9760 0.3092 1.1522 0.1856 0.3182

Metafrontier, CE∗ 0.8476 0.9791 0.3092 1.1568 0.2254

Table 9: Order-m efficiencies and metatechnology ratios, clusters based on sizeGroup Mean Median Minimum Maximum Std.Dev. % eff. obs.

Group efficiency, CEg 0.9011 1.0000 0.2428 1.6983 0.2008Cluster 1 Technology gap ratio 0.8972 0.9773 0.3659 1.1949 0.1541 0.1068

Metafrontier, CE∗ 0.8904 1.0000 0.2344 1.4207 0.2048

Group efficiency, CEg 0.9665 1.0000 0.3955 1.6893 0.1388Cluster 2 Technology gap ratio 0.9683 1.0000 0.5640 1.1364 0.0749 0.5979

Metafrontier, CE∗ 0.9417 1.0000 0.3772 1.1364 0.1343

Group efficiency, CEg 0.9788 1.0000 0.6376 1.0020 0.073Cluster 3 Technology gap ratio 0.9866 1.0000 0.7564 1.0000 0.0491 0.9014

Metafrontier, CE∗ 0.9787 1.0000 0.6376 1.0000 0.0730

22

Table 10: Distribution hypothesis tests (Li, 1996; Simarand Zelenyuk, 2006)

Efficiency distributions compared t-statistic p-value

Metafrontier vs. output mix clusters 3.4814 0.0002Metafrontier vs. environmental clusters 7.8215 0.0000Metafrontier vs. size clusters 2.7929 0.0026

cluster considered, the environmental conditions faced by the different municipalities play a remarkable

role, leading us to mislabel them as inefficient. Note also that the differences found between CEg and

CE∗ are irrespective of the number of observations in each cluster.

Figure 2 shows densities estimated using kernel smoothing of unconditioned and cluster-specific fron-

tiers. The tighter probability mass at unity shows that observations are indeed much closer to those in

their groups than to observations in other groups. These differences are also significant, as indicated by

the p-values in Table 10, which were obtained by applying the Simar and Zelenyuk (2006) test. The

substantial amount of probability mass found at the upper tail of the output mix distribution (Figure 2.a)

indicates that controlling for group membership contributes in a more modest way of explaining efficiency

differentials than in the case of clusters formed using environmental variables.

6. Concluding remarks

Over the last few years, a relevant area of research in the field of public economics and regional science

and urban economics has been the analysis of the efficiency of lower layers of government such as regional

governments or, as it is the case in this study, local governments. The topic is not only relevant per se, but

also because of recent events such as the economic and financial crisis. In some countries such as Spain,

economic activity has stalled, resulting in a sharp reduction of tax revenues and a simultaneous rapid

increase of public spending. Under these circumstances, the efficient management of public resources at

all levels of government becomes even more important.

Among the different levels of government, the literature devoted to the analysis of municipality effi-

ciency is now relatively large. However, although the number of theoretical and applied contributions is

high, there are still some critical issues which remain unsolved. One refers to the definition of municipality

output. This is a thorny question, especially if we take into account that, on the one hand, municipal-

ities face a budget constraint and, on the other, the law requires them to provide a minimum amount

of services and facilities. A related issue is the remarkably varied environmental conditions—in our case

defined as socioeconomic variables—that each municipality faces, which may have a marked effect on their

performance.

We deal with these issues using a two-stage procedure. In the first stage, municipalities are classified

into groups using cluster analysis taking into account variables based on their output mixes and envi-

ronmental conditions. We identify groups of municipalities that share some important features in these

fields, and the differences among them were statistically significant. In the second stage we assess how

23

Fig

ure

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ier

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,yea

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00

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Va

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Density

a)M

etaf

ront

ier

vs.

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ut

mix

0.5

1.0

1.5

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Va

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Density

b)

Met

afro

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.en

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24

results vary when considering that each municipality should be compared with those in its group rather

than with all municipalities.

The efficiency literature has dealt with the existence of groups and varying environmental conditions

following different criteria. Some authors (Battese and Rao, 2002; Battese et al., 2004; O’Donnell et al.,

2008) have proposed a methodology to compare the decision making units in different groups under the

assumption that technology differs across groups and, therefore, one should estimate both group frontiers

and a metafrontier. We deal with these concepts in the context of the order-m frontiers proposed by

Cazals et al. (2002), in order to tackle relevant problems such as the existence of outliers and the curse of

dimensionality. The severity of these problems, especially the latter, has not been fully acknowledged by

the efficiency literature (Simar and Wilson, 2008, p.441).

Our results indicate that both hypotheses are relevant, especially that referring to the relevance of

different environmental conditions—i.e., it is essential to control for the environment surrounding each

municipality. Although the literature on efficiency had proposed ways to control for them, our methods

are more robust to outliers and alleviate the curse of dimensionality. However, we must admit the groups

constructed, both using output mix and environmental variables, were partly subjective in terms of num-

ber of groups and composition because of the technique employed in their formation—cluster analysis.

Ideally, the research agenda should address how to define groups more objectively when natural boundaries

between them do not exist.

25

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